0
$\begingroup$

Let $\mu$ a finite Borel measure in $\mathbb{R}^d$ and $B_x$ an open disc with center $x$ and radius $1$.

Show that the function $f(x)=\mu (B_x)$ is measurable.

Could you give me some hints how I could do that??

$\endgroup$
2
$\begingroup$

It's more than measurable, it's lower semicontinuous: Pick a sequence $x_k\to x$, then $1_{B_{x}}\leq \liminf_k 1_{B_{x_k}}$. Therefore by Fatou's lemma $$ f(x)=\int_{B_x} d\mu \leq \int_{\mathbb{R}^n} \liminf_k 1_{B_{x_k}} d\mu \leq \liminf_k \int_{B_{x_k}} d\mu =\liminf_k f(x_k). $$

$\endgroup$
  • $\begingroup$ Could you explain to me how we get the inequality $$1_{B_x} \leq \lim inf_k 1_{B_{x_k}}$$ ?? $\endgroup$ – Mary Star Feb 9 '15 at 19:59
  • $\begingroup$ @MaryStar: If $y\in B_x$ then whenever we have $|x-x_k|<1-|x-y|$, we'll get $y\in B_{x_k}$. Arguing similarly if $y\notin \bar{B}_x$ we conclude that if $y\notin \partial B_x$ then $1_{B_{x_k}}(y)\to 1_{B_x}(y)$. But since, if $y\in \partial B_x$, $1_{B_x}(y)=0\leq 1_{B_{x_k}}(y)$, so the inequality follows. $\endgroup$ – Jose27 Feb 9 '15 at 22:33
  • $\begingroup$ Could you explain it further to me?? And also when we have shown that $f(x) \leq \lim inf_k f(x_k)$ why does this mean that $f$ is semicontinuous?? $\endgroup$ – Mary Star Feb 10 '15 at 2:38
  • $\begingroup$ What exactly are you not understanding? As for your other question see here. $\endgroup$ – Jose27 Feb 10 '15 at 2:41
  • $\begingroup$ Instead of showing that $f$ is semicontinuous, could we have also showed that $f$ is continuous?? Is the following correct?? When $x \rightarrow y$ we have that $f(x)-f(y) \rightarrow 0$ since the measure of the difference is the measure of the part of the balls that don't intersect. $\endgroup$ – Mary Star Feb 10 '15 at 2:46
0
$\begingroup$

Hint: Try $d=1$ first. Fubini's theorem might help.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.